Question

If u(x, y) = x²y + 3xy⁴, x = e^t and y = sin t, find `"du"/"dt"` and evaluate if at t = 0

Answer 1

u(x, y) = x²y + 3xy⁴, x = e^t, y = sin t

`"du"/("d"x) = "du"/("d"x) ("d"x)/"dt" + "du"/("d"y) (""y)/"dt"`

`"du"/("d"x) = 2xy + 3y^4, ("d"x)/"dt" = "e"^t`

`"du"/("d"y) = x^2 + 12xy^3, ("d"y)/"dt"` = cos t

∴ `"du"/"dt"` = (2xy + 3y⁴) e^t + (x² + 12xy³) cos t

= (2e^t sin t + 3 sin⁴t) e^t + (e^2t + 12 e^t sin³t) cos t

`"du"/"dt"` = e^t(2 e^t sin t + 3 sin⁴t + e^t cos t + 12 sin³t cos t)

At t = 0

`"du"/"dt"` = e⁰(2 e⁰ sin 0 + 3 sin⁴ 0 + e⁰ cos 0 + 12 sin³0 cos 0)

= 1(0 + 0 + 1 + 0) (cos (0) = 1, sin(0) = 0, e⁰ = 1)

`"du"/"dt"` = 1